Problem: All of the 3rd grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$2.00$ each for students, and the group paid $$42.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$19.50$ each for teachers and $$7.50$ each for students, and the group paid $$138.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+2y = 42}$ ${19.5x+7.5y = 138}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-19.5x-6y = -126}$ ${19.5x+7.5y = 138}$ Add the top and bottom equations together. $ 1.5y = 12 $ $ y = \dfrac{12}{1.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {6.5x+2y = 42}$ to find $x$ ${6.5x + 2}{(8)}{= 42}$ $6.5x+16 = 42$ $6.5x = 26$ $x = \dfrac{26}{6.5}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {19.5x+7.5y = 138}$ and get the same answer for $x$ ${19.5x + 7.5}{(8)}{= 138}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.